I agree it probably wasn't a good idea to define it as 'possessive;' although, while it does not mean literal possession it still has a general nuance of 'things that belong to' (i.e factors of) which is probably why there is no distinction in my mind between the two.

For example, in 人の住む所 has a stronger nuance of a place that people have made their own, that they choose to live in etc. rather than simply happening to live there.

It's more obvious in an expression like 人の言うことを聞かない

but I suppose it is helpful to illustrate the difference between this の and 私の犬

I believe what's going on here is that の and が were a lot more interchangeable in Classical Japanese. Eventually things settled to where の is usually possessive and が is always a subject marker, but の can still act like が in relative clauses. Incidentally, there are still rare vestiges of が as a possessive, such as the word 我が, meaning "our".

You can get a better feel for no and ga (and all the other particles) by continued listening and reading

*confused*

I tried to take a no + verb sentence and put it in a real sentence you might use. I mean saying "hito no taberu tokoro"(changed the verb of your example: "hito no sumu tokoro") isn't really useful in every day speech, so how would I put this clause in a sentence that can be used when talking to people? That's what I was trying to do with my example.

So using your example, "hito no sumu tokoro", how would I use this sentence to describe a specific place in mind? For example, let's say I want to say "this place" is "(a) place where people live". Or does changing the sentence in this way, change the grammar/particles of the no-sentence, and that's why you say to use "ga"?

I don't think there's anything wrong with の in the sentence you proposed. I agree が is -more common- in that sentence but the の is fine.I also would phrase it, レストランは人の食べるところです。 The ’が’ after レストラン seems off somehow. Otherwise, yes, that's how the grammar is applied, that's just not quite a natural place to apply it.

I don't think you would use this in everyday conversation much at all. It tends to be used in longer more complex sentences, so you're going to see it mostly in writing and in in-depth discussions. I'm sure there are some particular phrases where の is preferred, that are also used in everyday conversation but I can't think of them offhand.

I recommend you just keep in mind that sometimes の can act like が; if you read books much at all you'll see it plenty. (It's probably fortunate that the cases of が acting like の that furrykef mentioned appear exclusively in a few fixed phrases, so you don't really have to worry about that.)

This topic is very interesting to me. I never heard of の being interchangeable with が before.

I always figured in examples such as these that it was an implied noun. From Tae Kim's guide:

The 「の」 particle in this usage essentially replaces the noun and takes over the role as a noun itself. We can essentially treat adjectives and verbs just like nouns by adding the 「の」 particle to it. The particle then becomes a generic noun, which we can treat just like a regular noun.

白いのは、かわいい。 Thing that is white is cute. 授業に行くのを忘れた。 Forgot the event of going to class.

so using the original example:

神様のいない日曜日

-> 神様の「こと」 (が) いない日曜日

So basically the の nominalises so the end result is somewhat similar to のこと, and then the が is implied. This is how I always assumed it to be anyway.

I suppose you can think of it that way if it helps, but historically, の and が were essentially interchangeable (I'm sure they had some difference in tone or dialect or something, but I don't really study pre-reform Japanese so I don't know those details).

In modern Japanese, they became mostly differentiated, but の can still designate a subject sometimes. が cannot in general be used in place of の, but there are fixed expressions where が is still used even though の would normally be used.

Anyways, although it's buried under a slew of more familiar uses for の, you'll find this definition in the dictionary,